MATERIAL NONLINEAR ANALYSIS. using SolidWorks 2010 Simulation

Transcription

1 MATERIAL NONLINEAR ANALYSIS using SolidWorks 2010 Simulation

2 LM-ST-1 Learning Module Non-Linear Analysis What is a Learning Module? Title Page Guide A Learning Module (LM) is a structured, concise, and self-sufficient learning resource. An LM provides the learner with the required content in a precise and concise manner, enabling the learner to learn more efficiently and effectively. It has a number of characteristics that distinguish it from a traditional textbook or textbook chapter: An LM is learning objective driven, and its scope is clearly defined and bounded. The module is compact and precise in presentation, and its core material contains only contents essential for achieving the learning objectives. Since an LM is inherently concise, it can be learned relatively quickly and efficiently. An LM is independent and free-standing. Module-based learning is therefore nonsequential and flexible, and can be personalized with ease. Presenting the material in a contained and precise fashion will allow the user to learn effectively, reducing the time and effort spent and ultimately improving the learning experience. This is the first module on Non-linear analysis and covers a material nonlinearity study in FEM. It goes through all of the steps necessary to successfully complete an analysis, including geometry creation, material selection, boundary condition specification, meshing, solution, and validation. These steps are first covered conceptually and then worked through directly as they are applied to an example problem. Estimated Learning Time for This Module Estimated learning time for this LM is equivalent to three 50-minute lectures, or one week of study time for a 3 credit hour course. How to Use This Module The learning module is organized in sections. Each section contains a short explanation and a link to where that section can be found. The explanation will give you an idea of what content is in each section. The link will allow you to complete the parts of the module you are interested in, while being able to skip any parts that you might already be familiar with. The modularity of the LM allows for an efficient use of your time. 1

3 LM-ST-1 1. Learning Objectives The objective of this module is to introduce the user to the process of material non-linear analysis using FEM. Upon completion of the module, the user should have a good understanding of the necessary logical steps of an FEM analysis, and be able to perform the following tasks: Creating the solid geometry Assigning material properties Imposing displacement boundary conditions Applying external forces Meshing Running the analysis Verifying model correctness Processing needed results 2. Prerequisites In order to complete the learning module successfully, the following prerequisites are required: By subject area: o Statics; o Mechanics of Materials or Elasticity o Plasticity By topic: Knowledge of o force balance o statically equivalent force systems o elastic deformation o plastic deformation o Young s modulus o Tangent modulus o Plastic stress and strain o Poisson s ratio o displacement o strain o stress o von Mises stress o Yield criteria o Strain hardening o Work hardening o Plastic flow o Types of nonlinearities o Stress concentration factor o Tension and bending loading modes 2

4 LM-ST-1 3. Pre-test The pre-test should be taken before taking other sections of the module. The purpose of the pretest is to assess the user's prior knowledge in subject areas relevant to static structural analysis such as Mechanics of Materials. Questions are focused towards fundamental concepts including stress, strain, displacement, kinematic relationship, constitutive relationship, equilibrium, and material properties. 4. Tutorial Problem Statements A good tutorial problem should focus on the logical steps in FEM modeling and demonstrate as many aspects of the FEM software as possible. It should also be simple in mechanics with an analytical solution available for validation. Three tutorial problems are covered in this learning module. Tutorial Problem 1 A rectangular beam of 1 inch x 0.25 inch x 0.04 inch is restrained at one end as shown in the below figure. A Load of 315 lbf is applied as shown in the figure. The beam is made up of plain carbon steel (consider a Tangential modulus of 31e6 psi). Find the von-mises stress. Compare the FEM result with analytic result. Tutorial Problem 2 Figure 1: Tutorial Problem 1 A rectangular beam of 2 inches x 0.05 inches x 0.25 inches is fixed at one end as shown in the below figure. A Load of 10 lbf is applied at the other end vertically upwards. The beam is made up of plain carbon steel (consider a Tangential modulus of 31e6 psi). Find the von-mises stress. Find the plastic moment. 3

5 LM-ST-1 Figure 2: Tutorial problem 2 Tutorial Problem 3 A rectangular plate of 1 inch x 0.3 inches x inches with a hole of radius 0.05 inches at its center is fixed at one end as shown in the below figure. A Load of 100 lbf is applied as shown in the figure. The beam is made up of plain carbon steel (consider a Tangential modulus of 31e6 psi). Find the von-mises stress. Find the stress concentration factor. Figure 3: Tutorial problem 3 5. Conceptual Analysis Conceptual analysis is the abstraction of the logical steps in performing a task or solving a problem. Conceptual analysis for FEM simulation is problem type dependent but softwareindependent, and is fundamental in understanding and solving the problem. Conceptual analysis for static structural analysis reveals the following general logical steps: 4

6 LM-ST-1 1. Pre-processing o Geometry creation o Material property assignment o Restraint and applied load specification o Mesh generation 2. Solution 3. Post-processing 4. Validation 8. Post-test The post-test will be taken upon completion of the module. The first part of the post-test is from the pre-test to test knowledge gained by the user, and the second part is focused on the FEM simulation process covered by the tutorial. 9. Assessment The assessment is provided as a way to receive feedback about the module. The user evaluates several categories of the learning experience, including interactive learning, the module format, its effectiveness and efficiency, the appropriateness of the sections, and the overall learning experience. There is also the opportunity to give suggestions or comments about the module. 10. Practice Problems The user should be able to solve practice problems after taking the module. The practice problems provide a good reinforcement of the knowledge and skills learned in the module, and can be assigned as homework problems in teaching, or self study problems to enhance learning. These problems are similar to the tutorial problem worked in the module, but they involve different geometries and loading modes, stress concentration, and statically indeterminate beams. 5

7 Pre-test 1. When a stress in the material exceeds the yield stress, the material is said to be in : O plastic loading O elastic loading O fixed O Torsion 2. The basic fundamental principles of elasticity are applicable to plastic models O True O False 3. Which of the following are Yield criteria? O Mohr-Coulomb O Tresca O Drucker-Prager O All the above 4. Yield stress is dependent on O Temperature O size O strain rate O All the above 5. Which of these are the effects of work hardening? O decrease of density in material O formation of dislocations O elongation of material length O None of the above 6. Which of the following statements about von Mises stress is correct? O von Mises stress is a scalar. O von Mises stress is maximum along a principal direction. O von Mises stress has 6 independent components. O von Mises stress is equal to yield strength of the material. 6

8 7. Hooke s law holds good even after the material is loaded beyond its yield limit. O True O False 8. For a bar of uniform cross-section under axial loading in x direction, the Young s modulus is equal to O The ratio of the axial displacement to the axial normal stress O The ratio of the x-normal stress to the x-normal strain O The ratio of the xy-shear stress to the x-normal stress O The ratio of the xy-shear stress to the xy-shear strain 9. For a square plate of uniform thickness under unequal bi-axial loading in x and y directions, the shear modulus is equal to O The ratio of the x-normal stress to the x-normal strain O The ratio of the y-normal stress to the y-normal strain O The ratio of the xy-shear stress to the xy-shear strain O The ratio of the x-displacement to the x-normal stress 10. If the von Mises stress of material point A is equal to the von Mises stress of material point B, then each of the six stress components (i.e., xx, yy, zz, xy, yz, and zx ) of point A is equal to the corresponding stress component of point B. O True O False Conceptual Analysis Conceptual Analysis of Nonlinear analysis: Conceptual analysis for a static structural problem using finite element analysis reveals that the following logical steps and sub-steps are needed: 1. Pre-processing (building the model) 1. Geometry creation 2. Material property assignment 3. Boundary condition specification 7

9 o Prescribed displacement boundary condition (holding the model) o Applied force boundary condition (loading the model) 4. Mesh generation 2. Solution (running the simulation) 3. Post-processing (getting results) 4. Validation (checking) The above steps are explained in some detail as follows. 1. Pre-processing The pre-processing in FEM simulation is analogous to building the structure or making the specimen in physical testing. Several sub-steps involved in pre-processing are geometry creation, material property assignment, boundary condition specification, and mesh generation. The geometry of the structure to be analyzed is defined in the geometry creation step. After the solid geometry is created, the material properties of the solid are specified in the material property assignment step. The material required for the FEM analysis depends on the type of analysis. For example, in the elastic deformation analysis of an isotropic material under isothermal condition, only the modulus of elasticity and the Poisson s ratio are needed. For most novice users of FEM, the boundary condition specification step is probably the most challenging of all pre-processing steps. Two types of boundary conditions are possible. The first is prescribed displacement boundary condition which is analogous to holding or supporting the specimen in physical testing. The second is applied force boundary condition which is analogous to loading the specimen. Several factors contribute to the challenge of applying boundary conditions correctly: 1) Prescribed displacement boundary conditions expressed in terms such as u u const or const are mathematical simplifications, and boundarya x boundaryb frequently only represent supports in real structures approximately. As a result, choosing a good approximate mathematical representation can be a challenge. 2) How a boundary is restrained depends also on the element type. For example, for the "clamped" or "built-in" support, a boundary should be restrained as having zero nodal displacement if solid element is used, while for the same support, the boundary should be restrained as having zero nodal displacement and zero nodal rotation if shell element is used. 3) Frequently, the structure to be analyzed is not fully restrained from rigid body motion in the original problem statement. In order to obtain an FEM solution, auxiliary restraints become necessary. Over-restraining the model, however, leads to spurious stress results. The challenge is then adding auxiliary restraints to eliminate the possibility of rigid body motion without over-restraining the structure. 8

10 Because of the above challenges, one learning module will be devoted to boundary condition specification. Mesh generation is the process of discretizing the body into finite elements and assembling the discrete elements into an integral structure that approximates the original body. Most FEM packages have their own default meshing parameters to mesh the model and run the analysis while providing ways for the user to refine the mesh. 2. Solution The solution is the process of solving the governing equations resulting from the discretized FEM model. Although the mathematics for the solution process can be quite involved, this step is transparent to the user and is usually as simple as clicking a solution button or issuing the solution command. 3. Post-processing The purpose of an FEM analysis is to obtain wanted results, and this is what the postprocessing step is for. Typically, various components or measures of stress, strain, and displacement at any given location in the structure are available for putout. Additional quantities for output may include factory of safety, energy norm error, contact pressure, reaction force, strain energy density, etc. The way a quantity is outputted depends on the FEM software. 4. Validation Although validation is not a formal part of the FEM analysis, it is important to be included. Blindly trusting a simulation without checking its correctness can be dangerous. The validation usually involves comparing FEM results at one or more selected positions with exact or approximate solutions using classical approaches such as elasticity or mechanics of materials. Going through validation strengthens conceptual understanding and enhances learning. Conceptual Analysis of the Given Problem The goal of the FEM simulation is to analyze the behavior of the solid with the given forces acting on it. The problem shows a rectangular steel beam in the vertical plane which is fixed at one end and has a force of 315 lbf acting at the other end. The modulus of elasticity is 30 Mpsi and Poisson s ratio is Conceptual analysis of the current problem is described as follows. 1. Pre-processing (building the model) The geometry of the structure is first created using the design feature of the FEM package. Next, a material is assigned to the solid model. In the given problem, the 9

11 material of the beam is given as steel. Depending on the software, the material is either directly selected as steel from the material library, or the properties of the material given in the problem are inputted directly. After assigning the material properties, the boundary conditions are specified. The end that is attached to the wall will need a fixed restraint, which means zero displacement for all boundary nodes due to the solid mesh. The load is applied on the other end The next step is to mesh the solid to discretize it into finite elements. Generally, commercial FEA software has automatic default meshing parameters such as average element size of the mesh, quality of the mesh, etc. Here the default parameters provided by the software is used. 2. Solution (running the simulation) The next step is to run the simulation and obtain a solution. Usually the software provides several solver options. The default solver usually works well. For some problems, a particular solver may be faster or give more accurate results. 3. Post-processing (getting results) After the analysis is complete, the post-processing steps are performed. Results such as von Mises stress, various stresses, displacements, and strains can be viewed. 4. Validation (checking) Validation is the final step in the analysis process. In this step, the stresses acting on the beam are calculated by hand. These analytical solutions are compared with the software generated results to check the validity of the analysis. This completes the Conceptual Analysis section. Click the link below to continue with the learning module. SolidWorks-Specific FEM Tutorial 1 10

12 Overview: In this section, three tutorial problems will be solved using the commercial FEM software SolidWorks. Although the underlying principles and logical steps of an FEM simulation identified in the Conceptual Analysis section are independent of any particular FEM software, the realization of conceptual analysis steps will be software dependent. The SolidWorks-specific steps are described in this section. This is a step-by-step tutorial. However, it is designed such that those who are familiar with the details in a particular step can skip it and go directly into the next step. Tutorial Problem 1. A rectangular beam subjected to tensile loading 0. Launching SolidWorks SolidWorks Simulation is an integral part of the SolidWorks computer aided design software suite. The general user interface of SolidWorks is shown in Figure 8. Main menu Frequently used command icons Help icon Roll over to display File, Tools and other menus Figure 8: general user interface of SolidWorks. In order to perform FEM analysis, it is necessary to enable the FEM component, called SolidWorks Simulation, in the software. Step 1: Enabling SolidWorks Simulation 11

13 o Click "Tools" in the main menu. Select "Add-ins...". The Add-ins dialog window appears, as shown in Figure 9. o Check the boxes in both the Active Add-ins and Start Up columns corresponding to SolidWorks Simulation. o Checking the Active Add-ins box enables the SolidWorks for the current session. Checking the Start Up box enables the SolidWorks for all future sessions whenever SolidWorks starts up. 1. Pre-Processing Figure 9: Location of the SolidWorks icon and the boxes to be checked for adding it to the panel. Purpose: The purpose of pre-processing is to create an FEM model for use in the next step of the simulation, Solution. It consists of the following sub-steps: Geometry creation Material property assignment Boundary condition specification Mesh generation. 1.1 Geometry Creation Check SolidWorks Simulation boxes The purpose of Geometry Creation is to create a geometrical representation of the solid object or structure to be analyzed in FEM. In SolidWorks such a geometric model is called a part. In this tutorial, the necessary part has already been created in SolidWorks. The following steps will open up the part for use in the FEM analysis. Step 1: Opening the part for simulation. One of the following two options can be used. o Option1: Double click the following icon to open the embedded part file, tensile load.sldprt, in SolidWorks. 12

14 Click SolidWorks part file icon to open it ==> tensile load.sldprt o Option 2: Download the part file tensile load.sldprt from the web site Use the File menu in SolidWorks to open the downloaded part. The SolidWorks model tree will appear with the given part name at the top. Above the model tree, there should be various tabs labeled Features, Sketch, etc. If the Simulation tab is not visible, go back to steps 1 and 2 to enable the SolidWorks Simulation package. Step 2: Creating a Study o Click the Simulation tab above the model tree o Click on the drop down arrow under Study and select New Study as in Figure 10 o In the Name panel, give the study the name Tensile Loading o Select Nonlinear in the Type panel to study the Nonlinear behavior of the part under the load o In the options bar present at the bottom select static o Click OK to accept and close the menu Figure 10: The SolidWorks Study menu. 1.2 Material Property Assignment The Material Property Assignment sub-step assigns materials to different components of the part to be analyzed. All components must be assigned with appropriate material properties. Step 3: Opening the material property manager o In the upper left hand corner, click Apply Material. o The Material window appears as shown in Figure

15 Figure 11: The Material window. This will apply one material to all components. If the part is made of several components with different materials, open the model tree and apply this process to individual components. SolidWorks has a built-in material library that can be directly selected for the part. However, in this tutorial, material properties are defined using user input option. Step 4: Defining custom material properties o In the Material window, on the left panel under Select Material Source, select Custom defined o On the right panel, select Plasticity Von Mises in Model Type o Select English (IPS) under Units o Max von Mises Stress is selected for Default failure criterion o In the lower half of the panel, enter 30e6 for Elastic modulus, for Yield Strength, 31e6 for Tangent Modulus, 7.22e-006 for Thermal expansion coefficient, 0.28 for Poisson s ratio, for Mass density and 0 for Hardening factor. 1.3 Boundary Condition Specification In the Boundary Condition Specification sub-step, the restraints and loads on the part are defined. Here, the face of the beam attached to the wall needs to be restrained, and the force in the proper direction needs to be applied on the other end of the beam. Step 5: Opening the fixtures property manager o Right click on Fixtures in the model tree and select Advanced Fixtures 14

16 o Move the cursor into the graphic window. And constrain the part as shown in the below figures. Figure 12 (a): Applying an immovable restraint to the plate Figure 12 (b): Applying an immovable restraint to the beam. Once the desired face is visible, select the face on which to apply the restraint. Note that in the display panel, within the second box in the Type panel, Face<1> appears, 15

17 indicating that one surface is being selected. Clicking on this face in the graphics panel would deselect the face and select the other face for direction. Step 6: Restraining the member o Select the face as in Figure 13 o Once the face has been selected, click the green check mark to close the Fixture menu The next step is to load the beam with the applied force. The total force applied is 315 lbf acting towards right side of the beam as shown in the figure 6. Figure 13: Applying the tensile force. Step 7: Applying the Force o Right click on External Loads in the model tree and select Force. o Under the Force/Torque tab, click the Faces, Edges, Vertices, Reference Points for Force input field box to activate it, if not already active. o Click on the face on which the force is applied in the graphics window. Make sure the face is highlighted (turns blue) and appears in the input field box. o Choose Normal. Click the Face, Edge, Plane, Axis for Direction input field box to activate it. o Then click on Reverse direction to apply the force upwards. o Variation with time is left unchanged (linear). o Click OK to close the menu. 1.4 Mesh Generation 16

18 Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into elements. The mesh consists of a network of these elements. Step 9: Creating the mesh o Right click Mesh in the model tree and select Create mesh o Leave the mesh bar on its default value o Drop down the Advanced menu and make sure the mesh is high quality, not draft quality, by making sure the Draft Quality Mesh checkbox is not clicked which is shown in the figure 14. o Click OK to close the menu and generate the mesh. o Figure 15 shows the completed mesh Figure 14: Basic Mesh properties. 17

19 Figure 15: A completed mesh. Mesh Control in SolidWorks may be used to refine the mesh locally. The guiding principle is to refine mesh at locations of high stress gradient, such as regions around stress concentrators and locations of geometric changes. For the current problem, local mesh refinement is not pursued. 1.5 Setting the static study properties Purpose: The purpose of defining the static study properties is to define the iteration type and the parameters like the tolerances and the stepping options for solver and various other parameters. o Various parameters are defined as shown in the following figures 16 (a) & (b). 18

20 2. Solution Figures 16 (a) & 9 (b): Static study parameters. Purpose: The Solution is the step where the computer solves the simulation problem and generates results for use in the Post-Processing step. Step 1: Running the simulation o At the top of the screen, click Run o When the analysis is finished, the Results icon will appear on the model tree 3. Post-Processing Purpose: The purpose of the Post-Processing step is to process the results of interest. For this problem, the von Mises stress is the parameter of interest. From that we will calculate the bending moment of the beam. Step 1: Creating a stress plot o Right click Results on the model tree and select Define Stress Plot o Select von Mises as the stress type and psi as the unit o Unclick the Deformed Shape box and click OK to close the menu Figure 17 shows the resulting stress plot. 19

21 Figure 17: The von Mises stress plot. We can observe that the von mises stress that is obtained exceeds the yield strength of the material. This shows that the material is subjected to a plastic deformation. Step 2: Plotting the resultant deformation o The deformation due to the plastic loads can be plotted like the below figure Figure 18. Plot displaying the plastic deformation Step 3: Plotting plastic strain o To plot the plastic strain, click edit definition for strain plot. 20

22 o Then select plastic as shown in the figure 19. o Then the resulting figure can be obtained like in figure 20. o These results can now be compared to analytical solutions for validation. Figure 19. Selecting plastic strain Figure 20: Plot for plastic strain Note: If you want to view the stress at a particular point, right click on the stress plot and select Probe. Also in Chart Options, you can choose to display the minimum and maximum stress on the figure. 21

23 4. Validation Purpose: The purpose of the Validation step is to compare FEM solutions with analytical solutions, or known published results, to validate the correctness of the FEM model. For the current problem, closed form solutions based on elementary beam theory from Mechanics of Materials and plasticity are computed and compared with the FEM results. This will reveal whether or not the results of the finite element analysis are reasonable. In this problem, The beam is subjected to a tensile load. The stress equations from elementary beam theory are: Axial stress, α = = = psi α Elastic = psi ε Elastic = = = To find the plastic strain, we have considered a tangent modulus of 3.1e7 psi. So we can find the plastic stress at that particular tangent modulus and correspondingly we can find out the plastic moment value. α Plastic = x (Tangent Modulus) = ( ) x (3.1e7) = psi The above plastic stress is the value at that particular tangent modulus. Now we will consider the FEA value for plastic stress at the same Tangent Modulus. 22

24 From FEA: α Plastic = psi Stress Component SolidWorks Elementary Beam Theory % Error α psi psi 1.13 % The percent error is low for both stress components. The good agreement between the results using elementary beam theory and the FEM results validates the correctness of the FEM analysis. Tutorial Problem 2. A rectangular cantilever beam under a vertical bending force. 1. Pre-Processing Purpose: The purpose of pre-processing is to create an FEM model for use in the next step of the simulation, Solution. It consists of the following sub-steps: Geometry creation Material property assignment Boundary condition specification Mesh generation. 1.1 Geometry Creation The purpose of Geometry Creation is to create a geometrical representation of the solid object or structure to be analyzed in FEM. In SolidWorks such a geometric model is called a part. In this tutorial, the necessary part has already been created in SolidWorks. The following steps will open up the part for use in the FEM analysis. Step 1: Opening the part for simulation. One of the following two options can be used. o Option1: Double click the following icon to open the embedded part file, bend.sldprt, in SolidWorks. Click SolidWorks part file icon to open it ==> bend.sldprt 23

25 o Option 2: Download the part file bend.sldprt from the web site Use the File menu in SolidWorks to open the downloaded part. The SolidWorks model tree will appear with the given part name at the top. Above the model tree, there should be various tabs labeled Features, Sketch, etc. If the Simulation tab is not visible, go back to steps 1 and 2 to enable the SolidWorks Simulation package. Step 2: Creating a Study o Click the Simulation tab above the model tree o Click on the drop down arrow under Study and select New Study as in Figure 23 o In the Name panel, give the study the name Beambending o Select Nonlinear in the Type panel to study the Nonlinear behavior of the part under the load o In the options bar present at the bottom select static o Click OK to accept and close the menu Figure 23: The SolidWorks Study menu. 1.2 Material Property Assignment The Material Property Assignment substep assigns materials to different components of the part to be analyzed. All components must be assigned with appropriate material properties. Step 3: Opening the material property manager o In the upper left hand corner, click Apply Material. o The Material window appears as shown in Figure 24. Figure 24: The Material window. 24

26 This will apply one material to all components. If the part is made of several components with different materials, open the model tree and apply this process to individual components. SolidWorks has a built-in material library that can be directly selected for the part. However, in this tutorial, material properties are defined using user input option. Step 4: Defining custom material properties o In the Material window, on the left panel under Select Material Source, select Custom defined o On the right panel, select Plasticity Von Mises in Model Type o Select English (IPS) under Units o Max von Mises Stress is selected for Default failure criterion o In the lower half of the panel, enter 30e6 for Elastic modulus, for Yield Strength, 31e6 for Tangent Modulus, 7.22e-006 for Thermal expansion coefficient, 0.28 for Poisson s ratio, for Mass density and 0 for Hardening factor. 1.3 Boundary Condition Specification In the Boundary Condition Specification sub-step, the restraints and loads on the part are defined. Here, the face of the beam attached to the wall needs to be restrained, and the force in the proper direction needs to be applied on the other end of the beam. Step 5: Opening the fixtures property manager o Right click on Fixtures in the model tree and select Fixed Geometry o Move the cursor into the graphic window. As the cursor traverses the image of the model, notice a small icon accompany the cursor, and this icon change shapes when the cursor is at different locations. This indicates that the SolidWorks is in graphical selection mode, and different shapes indicate different identities would be selected: a square (icon) indicates the surface underneath the cursor will be selected if the mouse is clicked, a line (icon) for an edge or a line, and a dot (icon) for a point. In this tutorial problem, the entire end surface is restrained. 25

27 Figure 25: Applying an immovable restraint to the beam. At the initial orientation, however, the end to be restrained is not visible, and could not be selected. The model should be rotated to make the fixed end visible. To rotate the model either hold down the scroll bar and rotate with the mouse or change the orientation by clicking on the View Orientation icon in the top middle area of the workspace. Once the desired face is visible, select the face on which to apply the restraint. Note that in the display panel, within the second box in the Type panel, Face<1> appears, indicating that one surface is being selected. Clicking on this face in the graphics panel would deselect the face. Step 6: Restraining the member o Select the face as in Figure 25 o Once the face has been selected, click the green check mark to close the Fixture menu The next step is to load the beam with the applied force. The total force applied is 10 lbf acting vertically upwards as shown in the figure

28 Figure 26: Applying the bending force. Step 7: Applying the Force o Right click on External Loads in the model tree and select Force. o Under the Force/Torque tab, click the Faces, Edges, Vertices, Reference Points for Force input field box to activate it, if not already active. o Click on the face on which the force is applied in the graphics window. Make sure the face is highlighted (turns blue) and appears in the input field box. o Choose Selected direction instead of Normal. Click the Face, Edge, Plane, Axis for Direction input field box to activate it. o Then select the vertical edge of the face which is already selected for force to specify the direction of force. o Then click on Reverse direction to apply the force upwards. o Variation with time is left unchanged (linear). o Click OK to close the menu. 1.4 Mesh Generation Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into elements. The mesh consists of a network of these elements. Step 9: Creating the mesh o Right click Mesh in the model tree and select Create mesh o Leave the mesh bar on its default value o Drop down the Advanced menu and make sure the mesh is high quality, not draft 27

29 quality, by making sure the Draft Quality Mesh checkbox is not clicked which is shown in the figure 27. o Click OK to close the menu and generate the mesh. o Figure 28 shows the completed mesh Figure 27: Basic Mesh properties. Figure 28: A completed mesh. Mesh Control in SolidWorks may be used to refine the mesh locally. The guiding principle is to refine mesh at locations of high stress gradient, such as regions around stress concentrators and locations of geometric changes. For the current problem, local mesh refinement is not pursued. 28

30 1.5 Setting the static study properties Purpose: The purpose of defining the static study properties is to define the iteration type and the parameters like the tolerances and the stepping options for solver and various other parameters. o Various parameters are defined as shown in the following figures 29 (a) & (b). 2. Solution Figures 29 (a) & 29 (b): Static study parameters. Purpose: The Solution is the step where the computer solves the simulation problem and generates results for use in the Post-Processing step. Step 1: Running the simulation o At the top of the screen, click Run o When the analysis is finished, the Results icon will appear on the model tree 3. Post-Processing 29

31 Purpose: The purpose of the Post-Processing step is to process the results of interest. For this problem, the von Mises stress is the parameter of interest. From that we will calculate the bending moment of the beam. Step 1: Creating a stress plot o Right click Results on the model tree and select Define Stress Plot o Select von Mises as the stress type and psi as the unit o Unclick the Deformed Shape box and click OK to close the menu Figure 30 shows the resulting stress plot. Figure 30: The von Mises stress plot. We can observe that the von mises stress that is obtained exceeds the yield strength of the material. This shows that the material is subjected to a plastic deformation. Step 2: Plotting the resultant deformation o The deformation due to the plastic loads can be plotted like the below figure 30

32 Figure 31. Plot displaying the plastic deformation Step 3: Plotting plastic strain o To plot the plastic strain, click edit definition for strain plot. o Then select plastic as shown in the figure 32. o Then the resulting figure can be obtained like in figure 33. o These results can now be compared to analytical solutions for validation. Figure 32. Selecting plastic strain 31

33 Figure 33: Plot for plastic strain Note: If you want to view the stress at a particular point, right click on the stress plot and select Probe. Also in Chart Options, you can choose to display the minimum and maximum stress on the figure. 4. Validation Purpose: The purpose of the Validation step is to compare FEM solutions with analytical solutions, or known published results, to validate the correctness of the FEM model. For the current problem, closed form solutions based on elementary beam theory from Mechanics of Materials and plasticity are computed and compared with the FEM results. This will reveal whether or not the results of the finite element analysis are reasonable. In this problem, The beam is subjected to a bending load. The stress equations from elementary beam theory are: Moment M = (Force) x (length of the beam) = (10 lbf) x (2 inches) = 20 lbf-in 32

34 We know that, α Elastic = Where, M = Bending Moment C = distance from the neutral axis to the end of the beam I = Moment of inertia α Elastic = psi ε Elastic = = = To find the plastic strain, we have considered a tangent modulus of 3.1e7 psi. So we can find the plastic stress at that particular tangent modulus and correspondingly we can find out the plastic moment value. α Plastic = (ε Elastic ) x (Tangent Modulus) = ( ) x (3.1e7) = psi The above plastic stress is the value at that particular tangent modulus. Now we will consider the FEA value for plastic stress at the same Tangent Modulus. From FEA: α Plastic = psi Stress Component SolidWorks Elementary Beam Theory % Error α psi psi 6.7 % 33

35 For M>Me the beam is in the elastic-plastic regime: the core of the beam (between y=- c and y=c) is in the elastic regime, while the outer fibers are in the plastic regime ( = y). 34

36 = (39680 x 0.05 x ) / 4 = lb-in = (39680 x 0.05 x ) / 6 = lb-in Therefore, M p = 1.5 M e The above condition is also proved. The percent error is low for both stress components. The good agreement between the results using elementary beam theory and the FEM results validates the correctness of the FEM analysis. 35

37 Tutorial Problem 3. A rectangular plate with a hole at its center subjected to uniaxial loading. 1. Pre-Processing Purpose: The purpose of pre-processing is to create an FEM model for use in the next step of the simulation, Solution. It consists of the following sub-steps: Geometry creation Material property assignment Boundary condition specification Mesh generation. 1.1 Geometry Creation The purpose of Geometry Creation is to create a geometrical representation of the solid object or structure to be analyzed in FEM. In SolidWorks such a geometric model is called a part. In this tutorial, the necessary part has already been created in SolidWorks. The following steps will open up the part for use in the FEM analysis. Step 1: Opening the part for simulation. One of the following two options can be used. o Option1: Double click the following icon to open the embedded part file, L- plate.sldprt, in SolidWorks. Click SolidWorks part file icon to open it ==> plate.sldprt o Option 2: Download the part file plate.sldprt from the web site Use the File menu in SolidWorks to open the downloaded part. The SolidWorks model tree will appear with the given part name at the top. Above the model tree, there should be various tabs labeled Features, Sketch, etc. If the Simulation tab is not visible, go back to steps 1 and 2 to enable the SolidWorks Simulation package. Step 2: Creating a Study o Click the Simulation tab above the model tree o Click on the drop down arrow under Study and select New Study as in Figure 36 o In the Name panel, give the study the name Stressconcentration o Select Nonlinear in the Type panel to study the Nonlinear behavior of the part under the load 36

38 o In the options bar present at the bottom select static o Click OK to accept and close the menu 1.2 Material Property Assignment Figure 36: The SolidWorks Study menu. The Material Property Assignment sub-step assigns materials to different components of the part to be analyzed. All components must be assigned with appropriate material properties. Step 3: Opening the material property manager o In the upper left hand corner, click Apply Material. o The Material window appears as shown in Figure

39 Figure 37: The Material window. This will apply one material to all components. If the part is made of several components with different materials, open the model tree and apply this process to individual components. SolidWorks has a built-in material library that can be directly selected for the part. However, in this tutorial, material properties are defined using user input option. Step 4: Defining custom material properties o In the Material window, on the left panel under Select Material Source, select Custom defined o On the right panel, select Plasticity Von Mises in Model Type o Select English (IPS) under Units o Max von Mises Stress is selected for Default failure criterion o In the lower half of the panel, enter 30e6 for Elastic modulus, for Yield Strength, 31e6 for Tangent Modulus, 7.22e-006 for Thermal expansion coefficient, 0.28 for Poisson s ratio, for Mass density and 1/3 for Hardening factor. 38

40 1.3 Boundary Condition Specification In the Boundary Condition Specification sub-step, the restraints and loads on the part are defined. Here, the face of the beam attached to the wall needs to be restrained, and the force in the proper direction needs to be applied on the other end of the beam. Step 5: Opening the fixtures property manager o Right click on Fixtures in the model tree and select Fixed Geometry o Move the cursor into the graphic window. As the cursor traverses the image of the model, notice a small icon accompany the cursor, and this icon change shapes when the cursor is at different locations. This indicates that the SolidWorks is in graphical selection mode, and different shapes indicate different identities would be selected: a square (icon) indicates the surface underneath the cursor will be selected if the mouse is clicked, a line (icon) for an edge or a line, and a dot (icon) for a point. In this tutorial problem, the entire end surface is restrained. Figure 38: Applying an immovable restraint to the beam. At the initial orientation, however, the end to be restrained is not visible, and could not be selected. The model should be rotated to make the fixed end visible. To rotate the model either hold down the scroll bar and rotate with the mouse or change the orientation by clicking on the View Orientation icon in the top middle area of the workspace. Once the desired face is visible, select the face on which to apply the restraint. Note that in the display panel, within the second box in the Type panel, Face<1> appears, indicating that one surface is being selected. Clicking on this face in the graphics panel would deselect the face. 39

41 Step 6: Restraining the member o Select the face as in Figure 38 o Once the face has been selected, click the green check mark to close the Fixture menu The next step is to load the beam with the applied force. The total force applied is 100 lbf as shown in the figure 39. Figure 39: Applying the uniaxial tensile force. Step 7: Applying the Force o Right click on External Loads in the model tree and select Force. o Under the Force/Torque tab, click the Faces, Edges, Vertices, Reference Points for Force input field box to activate it, if not already active. o Click on the face on which the force is applied in the graphics window. Make sure the face is highlighted (turns blue) and appears in the input field box. o Choose Normal. Click the Face, Edge, Plane, Axis for Direction input field box to activate it. o Then click on Reverse direction to apply the force upwards. o Variation with time is left unchanged (linear). o Click OK to close the menu. 1.4 Mesh Generation Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into elements. The mesh consists of a network of these elements. 40

42 Step 9: Creating the mesh o Right click Mesh in the model tree and select Create mesh o Leave the mesh bar on its default value o Drop down the Advanced menu and make sure the mesh is high quality, not draft quality, by making sure the Draft Quality Mesh checkbox is not clicked which is shown in the figure 40. o Click OK to close the menu and generate the mesh. o Figure 41 shows the completed mesh Figure 40: Basic Mesh properties. Figure 41: A completed mesh. Mesh Control in SolidWorks may be used to refine the mesh locally. The guiding principle is to refine mesh at locations of high stress gradient, such as regions around 41

43 stress concentrators and locations of geometric changes. For the current problem, local mesh refinement is not pursued. 1.5 Setting the static study properties Purpose: The purpose of defining the static study properties is to define the iteration type and the parameters like the tolerances and the stepping options for solver and various other parameters. o Various parameters are defined as shown in the following figures 42 (a) & (b). 2. Solution Figures 42 (a) & 42 (b): Static study parameters. Purpose: The Solution is the step where the computer solves the simulation problem and generates results for use in the Post-Processing step. Step 1: Running the simulation o At the top of the screen, click Run o When the analysis is finished, the Results icon will appear on the model tree 42

44 3. Post-Processing Purpose: The purpose of the Post-Processing step is to process the results of interest. For this problem, the von Mises stress is the parameter of interest. From that we will calculate the bending moment of the beam. Step 1: Creating a stress plot o Right click Results on the model tree and select Define Stress Plot o Select von Mises as the stress type and psi as the unit o Unclick the Deformed Shape box and click OK to close the menu Figure 43 shows the resulting stress plot. Figure 43: The von Mises stress plot. We can observe that the von mises stress that is obtained exceeds the yield strength of the material. This shows that the material is subjected to a plastic deformation. Step 2: Plotting the resultant deformation o The deformation due to the plastic loads can be plotted like the below figure 43

45 Figure 44. Plot displaying the plastic deformation Step 3: Plotting plastic strain o To plot the plastic strain, click edit definition for strain plot. o Then select plastic as shown in the figure 45. o Then the resulting figure can be obtained like in figure 46. o These results can now be compared to analytical solutions for validation. Figure 45. Selecting plastic strain 44

46 Figure 46: Plot for plastic strain Note: If you want to view the stress at a particular point, right click on the stress plot and select Probe. Also in Chart Options, you can choose to display the minimum and maximum stress on the figure. 4. Validation Purpose: The purpose of the Validation step is to compare FEM solutions with analytical solutions, or known published results, to validate the correctness of the FEM model. For the current problem, closed form solutions based on elementary beam theory from Mechanics of Materials and plasticity are computed and compared with the FEM results. This will reveal whether or not the results of the finite element analysis are reasonable. In this problem, The beam is subjected to a axial tensile load. The stress equations from elementary beam theory are: Within elastic range: K t is defined by K t max nom 45

47 where the nominal stress The maximum normal stress F pwt pw, t is the thickness of the plate. Anet (w d)t w d is the x normal stress at the top of the hole. nom max When the deformation is plastic: For a nonhardening material, the stress concentration factor is ( ), whatever the yield criterion. When n = 1/3, the σ increases from ( ) to Y, the stress concentration factor decreases from 2 to a limiting value λ* = 1.35 for n = 1/3 and λ* = ( ) for other values of n. In the plastic region (r c), the Mises criterion is identically satisfied by writing σ r = sin Ø σ θ = cos ( ), where (r c), Where Ø is an auxiliary angle. At the end of the hole r = a, the boundary condition requires Ø = 0, while at the plastic boundary r = c, the continuity of the stresses requires Ø to have the value Ø c = sin -1 { ( ( ))} Here in this case, λ* = 1.35, then after calculations are done, = (Y) x (λ*) = (32) x (1.35) = 43.2 ksi Ø c = σ r = 43.2 ksi, here for this case it is uniaxial so σ θ = 0. 46

48 From FEA results: σ r = 36 ksi. For a hardening factor of 1/3. Stress Component SolidWorks Elementary Beam Theory % Error σ r 36 ksi 43.2 ksi 16.6 % 47

49 Attachment F. Assessment Post-test 1. The internal force per unit area acting inside the body when a force is applied on the body is called: O Stress O Strain O Displacement O Other 2. What is limiting point? 3. Stress concentration factor is the ratio of : 4. τ xy refers to: O nominal stress / maximum stress. O Elastic limit / stress acting. O Stress / strain. O Force / area O Shear stress in XY plane O Normal stress in X direction O Sum of normal stress in X and Y direction O Shear stress in YZ plane 5. What is perfect plasticity? 6. Bending moment induces: O Tensile stress O Compressive stress O Both tensile and compressive stress O Shear stress 7. What is von Mises stress? O The principle stress 48

50 Attachment F. Assessment O The normal stress O A scalar value that represents the total stress O A stress vector 8. What is Poisson s ratio? O The ratio of the contraction strain to the axial strain O The ratio of the shear stress to the normal stress O The ratio of the displacement to the normal stress O The ratio of shear stress to shear strain 9. What is Young s modulus? O The ratio of the normal stress to the normal strain O The ratio of the shear stress to the normal stress O The ratio of the displacement to the normal stress O The ratio of shear stress to shear strain 10. What is the shear modulus? O The ratio of the normal stress to the normal strain O The ratio of the shear stress to the normal stress O The ratio of the displacement to the normal stress O The ratio of shear stress to shear strain 11. What is Tangent Modulus? 12. What type of iterations is used in the non-linear analysis? 13. What is Geometric Nonlinearity? 14. For which type of materials, Von-Mises criterion is preferred? 15. What is the difference between elastic and plastic analysis? 49

51 Attachment F. Assessment Assessment 1. Do you feel it was bad to not have a teacher there to answer any questions you might have? O It didn t matter O It would have been nice O I really wanted to ask a question 2. How did the interactivity of the program affect your learning? O Improved it a lot O Improved it some O No difference O Hurt it some O Hurt it a lot 3. The six levels of Bloom s Taxonomy are listed below. Rank how well this learning module covers each level. 5 meaning exceptionally well and 1 meaning very poor. 1. Knowledge (remembering previously learned material) O 5 O 4 O 3 O 2 O 1 2. Comprehension (the ability to grasp the meaning of the material and give examples) O 5 O 4 O 3 O 2 O 1 3. Application (the ability to use the material in new situations) O 5 O 4 O 3 O 2 O 1 50

52 Attachment F. Assessment 4. Analysis (the ability to break down material into its component parts so that its organizational structure may be understood) O 5 O 4 O 3 O 2 O 1 5. Synthesis (the ability to put parts together to form a new whole) O 5 O 4 O 3 O 2 O 1 6. Evaluation (the ability to judge the value of the material for a given purpose) O 5 O 4 O 3 O 2 O 1 4. Do you think the mixed text and video format works well? O Yes O Indifferent O No 5. Do you think the module presents an affective method of learning FEA? O Yes O Indifferent O No 6. Did you prefer this module over the traditional classroom learning experience? Why or why not. 51

53 Attachment F. Assessment 7. How accurate would it be to call this module self-contained and stand-alone? O Very accurate O Accurate O Indifferent O Inaccurate O Very inaccurate 8. What specifically did you like and/or dislike about the module. 9. How useful were the practice problems? O Very helpful O Helpful O Indifferent O Unhelpful O Very unhelpful 10. Was there any part of the module that you felt was unnecessary of redundant? Was there a need for any additional parts? 11. Please list any suggestions for improving this module. 12. Overall, how would you rate your experience taking this module? O Excellent O Fair O Average O Poor O Awful 52

54 Attachment G. Practice Problems Practice Problems 1. A rectangular beam of 1 inch x 0.25 inch x 0.04 inch is restrained at one end as shown in the below figure. A Load of 315 lbf is applied as shown in the figure. The beam is made up of plain carbon steel (consider a Tangential modulus of 33e6 psi). Find the von-mises stress. Compare the FEM result with analytic result. 2. A circular beam of 1 inch length and 0.05 inch diameter is restrained at one end as shown in the below figure. A Load of 100 lbf is applied as shown in the figure. The beam is made up of plain carbon steel (consider a Tangential modulus of 31e6 psi). Find the von-mises stress. Compare the FEM result with analytic result. 3. A rectangular beam of 2 inches x 0.05 inches x 0.25 inches is fixed at one end as shown in the below figure. A Load of 10 lbf is applied at the other end vertically upwards. The beam is made up of plain carbon steel (consider a Tangential modulus of 33e6 psi). Find the von-mises stress. Find the plastic moment. 53